#8831331

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

(3,4)(6,7)(10,12)(11,17)(16,20)(18,23)(27,33)(30,34)(35,39)(38,40);(1,2,5,8)(3,6,4,7)(9,10,14,12)(11,15,17,19)(13,20,21,16)(18,24,23,22)(25,26)(27,30)(28,36)(29,32)(31,37)(33,34)(35,40)(38,39);(1,3,5,4)(2,6,8,7)(11,17)(15,19)(18,23)(22,24)(25,26,29,32)(27,30,33,34)(28,31,37,36)(35,38,39,40);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,24)(21,23)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,37)(38,39);(1,2)(3,6)(4,7)(5,8)(9,12,14,10)(11,15,17,19)(13,20,21,16)(18,22,23,24)(25,27)(26,30)(28,35)(29,33)(31,38)(32,34)(36,40)(37,39);(1,4,5,3)(2,7,8,6)(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8738527);